I don’t agree with James S-C’s summary, though it’s correct for certain products. For example, in the US, car manufacturers offer “warranties”. In practice, the two words are often used interchangeably. In general a warranty is more specific - you can think of it as a piece of paper spelling out specific promises made by the seller of a product. “Guarantee” isn’t limited to product sales and is more conceptual. For example, a parent might “guarantee” a loan that his child takes out. An obligation by itself is a requirement to do something, regardless of the reason. “Responsibility” is tied to a moral or ethical reason for doing something. But they often get combined with other words, so you can talk of “moral obligation” or “legal responsibility”.

James answered the guarantee vs warrantee question. Let me answer the second half. Responsibility is less strong than obligation. Obligation can imply an outside force. Obligation is a moral or legal duty. Responsibility is not a legal term. Responsibility is also a duty, but often one you choose to take upon your own shoulders.

A guarantee is something given by the manufacturer. A warranty is given by the retailer. Walmart sells computers. The manufacturer, say HP, guarantees the computer is free from defects. If there is a defect, it can be returned. Walmart is willing to sell you a warranty that will cover a certain number of months. If a problem arises during that time, you can bring it to Walmart to get fixed or replaced.